// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2010 Manuel Yguel <manuel.yguel@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_POLYNOMIAL_SOLVER_H
#define EIGEN_POLYNOMIAL_SOLVER_H

namespace Eigen {

/** \ingroup Polynomials_Module
 *  \class PolynomialSolverBase.
 *
 * \brief Defined to be inherited by polynomial solvers: it provides
 * convenient methods such as
 *  - real roots,
 *  - greatest, smallest complex roots,
 *  - real roots with greatest, smallest absolute real value,
 *  - greatest, smallest real roots.
 *
 * It stores the set of roots as a vector of complexes.
 *
 */
template <typename _Scalar, int _Deg> class PolynomialSolverBase
{
public:
    EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar, _Deg == Dynamic ? Dynamic : _Deg)

    typedef _Scalar Scalar;
    typedef typename NumTraits<Scalar>::Real RealScalar;
    typedef std::complex<RealScalar> RootType;
    typedef Matrix<RootType, _Deg, 1> RootsType;

    typedef DenseIndex Index;

protected:
    template <typename OtherPolynomial> inline void setPolynomial(const OtherPolynomial& poly) { m_roots.resize(poly.size() - 1); }

public:
    template <typename OtherPolynomial> inline PolynomialSolverBase(const OtherPolynomial& poly) { setPolynomial(poly()); }

    inline PolynomialSolverBase() {}

public:
    /** \returns the complex roots of the polynomial */
    inline const RootsType& roots() const { return m_roots; }

public:
    /** Clear and fills the back insertion sequence with the real roots of the polynomial
     * i.e. the real part of the complex roots that have an imaginary part which
     * absolute value is smaller than absImaginaryThreshold.
     * absImaginaryThreshold takes the dummy_precision associated
     * with the _Scalar template parameter of the PolynomialSolver class as the default value.
     *
     * \param[out] bi_seq : the back insertion sequence (stl concept)
     * \param[in]  absImaginaryThreshold : the maximum bound of the imaginary part of a complex
     *  number that is considered as real.
     * */
    template <typename Stl_back_insertion_sequence>
    inline void realRoots(Stl_back_insertion_sequence& bi_seq, const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const
    {
        using std::abs;
        bi_seq.clear();
        for (Index i = 0; i < m_roots.size(); ++i)
        {
            if (abs(m_roots[i].imag()) < absImaginaryThreshold)
            {
                bi_seq.push_back(m_roots[i].real());
            }
        }
    }

protected:
    template <typename squaredNormBinaryPredicate> inline const RootType& selectComplexRoot_withRespectToNorm(squaredNormBinaryPredicate& pred) const
    {
        Index res = 0;
        RealScalar norm2 = numext::abs2(m_roots[0]);
        for (Index i = 1; i < m_roots.size(); ++i)
        {
            const RealScalar currNorm2 = numext::abs2(m_roots[i]);
            if (pred(currNorm2, norm2))
            {
                res = i;
                norm2 = currNorm2;
            }
        }
        return m_roots[res];
    }

public:
    /**
     * \returns the complex root with greatest norm.
     */
    inline const RootType& greatestRoot() const
    {
        std::greater<RealScalar> greater;
        return selectComplexRoot_withRespectToNorm(greater);
    }

    /**
     * \returns the complex root with smallest norm.
     */
    inline const RootType& smallestRoot() const
    {
        std::less<RealScalar> less;
        return selectComplexRoot_withRespectToNorm(less);
    }

protected:
    template <typename squaredRealPartBinaryPredicate>
    inline const RealScalar& selectRealRoot_withRespectToAbsRealPart(squaredRealPartBinaryPredicate& pred,
                                                                     bool& hasArealRoot,
                                                                     const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const
    {
        using std::abs;
        hasArealRoot = false;
        Index res = 0;
        RealScalar abs2(0);

        for (Index i = 0; i < m_roots.size(); ++i)
        {
            if (abs(m_roots[i].imag()) <= absImaginaryThreshold)
            {
                if (!hasArealRoot)
                {
                    hasArealRoot = true;
                    res = i;
                    abs2 = m_roots[i].real() * m_roots[i].real();
                }
                else
                {
                    const RealScalar currAbs2 = m_roots[i].real() * m_roots[i].real();
                    if (pred(currAbs2, abs2))
                    {
                        abs2 = currAbs2;
                        res = i;
                    }
                }
            }
            else if (!hasArealRoot)
            {
                if (abs(m_roots[i].imag()) < abs(m_roots[res].imag()))
                {
                    res = i;
                }
            }
        }
        return numext::real_ref(m_roots[res]);
    }

    template <typename RealPartBinaryPredicate>
    inline const RealScalar& selectRealRoot_withRespectToRealPart(RealPartBinaryPredicate& pred,
                                                                  bool& hasArealRoot,
                                                                  const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const
    {
        using std::abs;
        hasArealRoot = false;
        Index res = 0;
        RealScalar val(0);

        for (Index i = 0; i < m_roots.size(); ++i)
        {
            if (abs(m_roots[i].imag()) <= absImaginaryThreshold)
            {
                if (!hasArealRoot)
                {
                    hasArealRoot = true;
                    res = i;
                    val = m_roots[i].real();
                }
                else
                {
                    const RealScalar curr = m_roots[i].real();
                    if (pred(curr, val))
                    {
                        val = curr;
                        res = i;
                    }
                }
            }
            else
            {
                if (abs(m_roots[i].imag()) < abs(m_roots[res].imag()))
                {
                    res = i;
                }
            }
        }
        return numext::real_ref(m_roots[res]);
    }

public:
    /**
     * \returns a real root with greatest absolute magnitude.
     * A real root is defined as the real part of a complex root with absolute imaginary
     * part smallest than absImaginaryThreshold.
     * absImaginaryThreshold takes the dummy_precision associated
     * with the _Scalar template parameter of the PolynomialSolver class as the default value.
     * If no real root is found the boolean hasArealRoot is set to false and the real part of
     * the root with smallest absolute imaginary part is returned instead.
     *
     * \param[out] hasArealRoot : boolean true if a real root is found according to the
     *  absImaginaryThreshold criterion, false otherwise.
     * \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
     *  whether or not a root is real.
     */
    inline const RealScalar& absGreatestRealRoot(bool& hasArealRoot, const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const
    {
        std::greater<RealScalar> greater;
        return selectRealRoot_withRespectToAbsRealPart(greater, hasArealRoot, absImaginaryThreshold);
    }

    /**
     * \returns a real root with smallest absolute magnitude.
     * A real root is defined as the real part of a complex root with absolute imaginary
     * part smallest than absImaginaryThreshold.
     * absImaginaryThreshold takes the dummy_precision associated
     * with the _Scalar template parameter of the PolynomialSolver class as the default value.
     * If no real root is found the boolean hasArealRoot is set to false and the real part of
     * the root with smallest absolute imaginary part is returned instead.
     *
     * \param[out] hasArealRoot : boolean true if a real root is found according to the
     *  absImaginaryThreshold criterion, false otherwise.
     * \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
     *  whether or not a root is real.
     */
    inline const RealScalar& absSmallestRealRoot(bool& hasArealRoot, const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const
    {
        std::less<RealScalar> less;
        return selectRealRoot_withRespectToAbsRealPart(less, hasArealRoot, absImaginaryThreshold);
    }

    /**
     * \returns the real root with greatest value.
     * A real root is defined as the real part of a complex root with absolute imaginary
     * part smallest than absImaginaryThreshold.
     * absImaginaryThreshold takes the dummy_precision associated
     * with the _Scalar template parameter of the PolynomialSolver class as the default value.
     * If no real root is found the boolean hasArealRoot is set to false and the real part of
     * the root with smallest absolute imaginary part is returned instead.
     *
     * \param[out] hasArealRoot : boolean true if a real root is found according to the
     *  absImaginaryThreshold criterion, false otherwise.
     * \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
     *  whether or not a root is real.
     */
    inline const RealScalar& greatestRealRoot(bool& hasArealRoot, const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const
    {
        std::greater<RealScalar> greater;
        return selectRealRoot_withRespectToRealPart(greater, hasArealRoot, absImaginaryThreshold);
    }

    /**
     * \returns the real root with smallest value.
     * A real root is defined as the real part of a complex root with absolute imaginary
     * part smallest than absImaginaryThreshold.
     * absImaginaryThreshold takes the dummy_precision associated
     * with the _Scalar template parameter of the PolynomialSolver class as the default value.
     * If no real root is found the boolean hasArealRoot is set to false and the real part of
     * the root with smallest absolute imaginary part is returned instead.
     *
     * \param[out] hasArealRoot : boolean true if a real root is found according to the
     *  absImaginaryThreshold criterion, false otherwise.
     * \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
     *  whether or not a root is real.
     */
    inline const RealScalar& smallestRealRoot(bool& hasArealRoot, const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision()) const
    {
        std::less<RealScalar> less;
        return selectRealRoot_withRespectToRealPart(less, hasArealRoot, absImaginaryThreshold);
    }

protected:
    RootsType m_roots;
};

#define EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES(BASE) \
    typedef typename BASE::Scalar Scalar;                  \
    typedef typename BASE::RealScalar RealScalar;          \
    typedef typename BASE::RootType RootType;              \
    typedef typename BASE::RootsType RootsType;

/** \ingroup Polynomials_Module
  *
  * \class PolynomialSolver
  *
  * \brief A polynomial solver
  *
  * Computes the complex roots of a real polynomial.
  *
  * \param _Scalar the scalar type, i.e., the type of the polynomial coefficients
  * \param _Deg the degree of the polynomial, can be a compile time value or Dynamic.
  *             Notice that the number of polynomial coefficients is _Deg+1.
  *
  * This class implements a polynomial solver and provides convenient methods such as
  * - real roots,
  * - greatest, smallest complex roots,
  * - real roots with greatest, smallest absolute real value.
  * - greatest, smallest real roots.
  *
  * WARNING: this polynomial solver is experimental, part of the unsupported Eigen modules.
  *
  *
  * Currently a QR algorithm is used to compute the eigenvalues of the companion matrix of
  * the polynomial to compute its roots.
  * This supposes that the complex moduli of the roots are all distinct: e.g. there should
  * be no multiple roots or conjugate roots for instance.
  * With 32bit (float) floating types this problem shows up frequently.
  * However, almost always, correct accuracy is reached even in these cases for 64bit
  * (double) floating types and small polynomial degree (<20).
  */
template <typename _Scalar, int _Deg> class PolynomialSolver : public PolynomialSolverBase<_Scalar, _Deg>
{
public:
    EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar, _Deg == Dynamic ? Dynamic : _Deg)

    typedef PolynomialSolverBase<_Scalar, _Deg> PS_Base;
    EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES(PS_Base)

    typedef Matrix<Scalar, _Deg, _Deg> CompanionMatrixType;
    typedef typename internal::conditional<NumTraits<Scalar>::IsComplex, ComplexEigenSolver<CompanionMatrixType>, EigenSolver<CompanionMatrixType>>::type
        EigenSolverType;
    typedef typename internal::conditional<NumTraits<Scalar>::IsComplex, Scalar, std::complex<Scalar>>::type ComplexScalar;

public:
    /** Computes the complex roots of a new polynomial. */
    template <typename OtherPolynomial> void compute(const OtherPolynomial& poly)
    {
        eigen_assert(Scalar(0) != poly[poly.size() - 1]);
        eigen_assert(poly.size() > 1);
        if (poly.size() > 2)
        {
            internal::companion<Scalar, _Deg> companion(poly);
            companion.balance();
            m_eigenSolver.compute(companion.denseMatrix());
            m_roots = m_eigenSolver.eigenvalues();
            // cleanup noise in imaginary part of real roots:
            // if the imaginary part is rather small compared to the real part
            // and that cancelling the imaginary part yield a smaller evaluation,
            // then it's safe to keep the real part only.
            RealScalar coarse_prec = RealScalar(std::pow(4, poly.size() + 1)) * NumTraits<RealScalar>::epsilon();
            for (Index i = 0; i < m_roots.size(); ++i)
            {
                if (internal::isMuchSmallerThan(numext::abs(numext::imag(m_roots[i])), numext::abs(numext::real(m_roots[i])), coarse_prec))
                {
                    ComplexScalar as_real_root = ComplexScalar(numext::real(m_roots[i]));
                    if (numext::abs(poly_eval(poly, as_real_root)) <= numext::abs(poly_eval(poly, m_roots[i])))
                    {
                        m_roots[i] = as_real_root;
                    }
                }
            }
        }
        else if (poly.size() == 2)
        {
            m_roots.resize(1);
            m_roots[0] = -poly[0] / poly[1];
        }
    }

public:
    template <typename OtherPolynomial> inline PolynomialSolver(const OtherPolynomial& poly) { compute(poly); }

    inline PolynomialSolver() {}

protected:
    using PS_Base::m_roots;
    EigenSolverType m_eigenSolver;
};

template <typename _Scalar> class PolynomialSolver<_Scalar, 1> : public PolynomialSolverBase<_Scalar, 1>
{
public:
    typedef PolynomialSolverBase<_Scalar, 1> PS_Base;
    EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES(PS_Base)

public:
    /** Computes the complex roots of a new polynomial. */
    template <typename OtherPolynomial> void compute(const OtherPolynomial& poly)
    {
        eigen_assert(poly.size() == 2);
        eigen_assert(Scalar(0) != poly[1]);
        m_roots[0] = -poly[0] / poly[1];
    }

public:
    template <typename OtherPolynomial> inline PolynomialSolver(const OtherPolynomial& poly) { compute(poly); }

    inline PolynomialSolver() {}

protected:
    using PS_Base::m_roots;
};

}  // end namespace Eigen

#endif  // EIGEN_POLYNOMIAL_SOLVER_H
